I know both countries operated in unison during the conflict, however, were there any instances of US and ANZAC line units fighting together in specific battles?
Early in the Australian deployment to Vietnam, the 1st Battalion, Royal Australian Regiment (1 RAR) operated with US forces. In Operation Hump (November 1965) and Operation Crimp (January 1966), 1 RAR was attached to the US 173rd Airborne Brigade. During Operation Hump, 1 RAR fought a battle alongside the US 1st Battalion, 503rd Infantry Regiment (1/503), although the Dong Nai River (Song Dong Nai) separated the Australian and US forces, and their two battles remained separate.
In Operation Crimp, 1 RAR fought alongside 5 US battalions to locate and destroy a major Viet Cong headquarters and tunnel complex. The operation was partly successful, but the resources available were far from sufficient to deal with the 200km of tunnels. 1 RAR played a key role, due to the Viet Cong headquarters being in their area of operation.
In the Battle of Suoi Bong Trang, February 1966, 1 RAR, supported by a New Zealand artillery battery, fought attached to the US 1st Division.
Following this, the Australian (and New Zealand) forces were separated from the US forces, and with a second Australian battalion (and later also a third) formed the 1st Australian Task Force (1 ATF), in Phuoc Tuy Province. New Zealand infantry were deployed in Vietnam in 1967, and were integrated with the Australian forces to form ANZAC battalions (so in a technical sense, ANZAC line units only formed after closely-integrated operation with US forces had ceased)
The separation of US and Australian forces was partly driven by differences in tactical/operational doctrine between the US and Australian forces complicating working together, and partly driven by an increased Australian force making independent operation feasible.
Australian and ANZAC forces still operated with US forces at times. One notable battle was the post-Tet two and a half month Battle of Hat Dich, December 1968 to February 1969, when 1 ATF fought alongside US, South Vietnamese and Thai forces. This was a long and sprawling battle, spread over 50km from north to south, and the different forces had their own areas of operation, so cooperation was not as close as when 1 RAR fought attached to the US 173rd Airborne and 1st Division. Still, US and Thai artillery supported Australian/ANZAC and ARVN forces, and Australian/ANZAC ground forces fought alongside US and ARVN ground forces on occasion during the battle.